Clique-Transversal Sets in Cubic Graphs
Clique-Transversal Sets in Cubic Graphs Zuosong Liang1 , Erfang Shan1,2, , and T.C.E. Cheng2 1
Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China [email protected] 2 Department of Logistics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong [email protected]
Abstract. A clique-transversal set S of a graph G is a set of vertices of G such that S meets all cliques of G. The clique-transversal number, denoted τc (G), is the minimum cardinality of a clique-transversal set in G. In this paper we present an upper bound and a lower bound on τc (G) for cubic graphs, and characterize the extremal cubic graphs achieving the lower bound. In addition, we present a sharp upper bound on τc (G) for claw-free cubic graphs. Keywords: Clique-transversal number; Cubic graph; Claw-free; Bound.
1
Introduction
All graphs considered here are finite, simple and nonempty. For standard graph theory terminology not given here we refer the reader to [5]. Let G = (V, E) be a graph with vertex set V and edge set E. For a vertex v ∈ V , the degree of v is denoted by d(v) and a vertex of degree 0 is said to be an isolated vertex. If d(v) = k for all v ∈ V , then we call G k-regular. In particular, a 3-regular graph is also called a cubic graph. For a subset S ⊆ V , the subgraph induced by S is denoted by G[S], and let dS (v) denote the number of vertices in S that are adjacent to v. For two disjoint subsets T and S of V , write e[T, S] for the number of edges between T and S. The matching number of G is the maximum cardinality among the independent sets of edges of G and is denoted by α1 (G). A perfect matching in G is a matching with the property that every vertex in G is incident with an edge of the matching. A set U ⊆ V is called a vertex cover of G if every edge of G is incident with a vertex in U . The covering number, denoted by α0 (G), is the minimum cardinality of a vertex cover of G. A subset S of V is called a dominating set if every vertex of V − S is adjacent to some vertex in S. The domination
This research was partially supported by The Hong Kong Polytechnic University under grant number G-YX69, the National Nature Science Foundation of China under grant 10571117, the ShuGuang Plan of Shanghai Education Development Foundation under grant 06SG42 and the Development Foundation of Shanghai Education Committee under grant 05AZ04. Corresponding author.
B. Chen, M. Paterson, and G. Zhang (Eds.): ESCAPE 2007, LNCS 4614, pp. 107–115, 2007. c Springer-Verlag Berlin Heidelberg 2007
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number γ(G) of G is the minimum cardinality taken over all dominating sets of G. Domination in graphs has been well studied (see [12]). The concept of the clique-transversal set in graphs can be regarded as a special case of the transversal set in hypergraph theory, which is closely related to domination and seems to have been introduced in [1]. A clique C of a graph G is a complete subgraph maximal under inclusion and |C| ≥ 2. A set D ⊆ V in G is called a clique-transversal set if for every clique C of G, D ∩ V (C) = ∅. The clique-transversal number, denoted τc (G), is the minimum cardinality of a clique-transversal set of G. By definitions, each clique-transversal set in G is clearly a dominating set, so γ(G) ≤ τc (G). To motivate the study of the clique-transversal set in graphs, we present two examples of application where this concept may be used. An application is in terms of communication networks. Consider a graph associated with a communication network where the vertices in the graph correspond to the sites of the network, a clique usually represents a cluster of sites that has the best possible ability to rapidly exchange information among the members of the cluster. The clique-transversal set in the graph is faster to control all clusters and keeps the ability of dominating the whole network. Another application may be found in social networks theory. Every vertex of a graph represents an actor and an edge represents a relationship between two actors. A clique can be viewed as a maximal group of members that have the same property, while a clique-transversal set can be regarded as some kind of organization in the social networks. Then a clique-transversal set claims that each clique in the social networks owns at least one position in this organization. Erd˝ os, Gallai and Tuza [10] observed that the problem of finding a minimum clique-transversal set for an arbitrary graph is NP-hard. Further, it has been proved that the problem is still NP-hard on split graphs (a subclass of chordal graphs) [7], cocomparability, planar, line and total graphs [11], undirected path graphs, and k-trees with unbounded k [6]. However, there are polynomial time algorithms to find τc for comparability graphs [4], strongly chordal graphs [7], Helly circular-arc graphs [16], and distance-hereditary graphs [14]. In [10], Erd˝ os et al. investigated the bounds on τc , and√showed that every graph of order n has clique-transversal number at most n − 2n + 32 , and if all cliques are relatively large, then a sightly better upper bound can be obtained. However, they also observed that τc (G) can be very close to n = |V (G)|, namely τc = n − o(n) can hold. It is interesting to note that τc drastically decreases when some assumptions are put on the graph G. From this point of view, Tuza [17] and Andreae [2] established upper bounds on τc for chordal graphs. Andreae et al. [3] studied classes of graphs G of order n for which τc (G) ≤ n/2. They showed that (i) all connected line graphs with the exception of odd cycles, and (ii) all complements of line graphs with the exception of five small graphs have clique-transversal numbers at most one-half their orders. For other investigations on the cliquetransversal number of graphs, we refer the reader to [6, 8, 9, 13, 15, 18]. In this paper we shall focus on cubic graphs. We show that if G is any cubic graph, then 5n/14 ≤ τc (G) ≤ 2n/3, and the extremal graphs attaining the lower
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bound are characterized. Also, we show that a claw-free cubic graph has cliquetransversal number at most one-half its order, and the upper bound is sharp.
2
Clique-Transversal Number in Cubic Graphs
In this section we give lower and upper bounds on the clique-transversal number of a cubic graph in terms of its order and we characterize the graph attaining the lower bound. For this purpose, we define a family F of graphs as follows: For each integer k ≥ 1, let J0 be the graph obtained from a complete graph K4 on four vertices by deleting one edge, and let J be the disjoint union of J1 , J2 , . . . , J3k of 3k copies of J0 . Let Fk be a family of cubic graphs obtained from J by adding 2k new vertices and 6k edges that join each new vertex exactly to three vertices of degree 2 of J so that each vertex has degree 3. Let F = {Fk | k ≥ 1}. The graph F1 is shown in Fig. 1. For notational convenience, every clique of order m of a graph G is called a Km -clique of G, and a component of G is called a H-component if it is isomorphic to a given graph H. Theorem 1. If G is a cubic graph of order n ≥ 5, then 2 5 n ≤ τc (G) ≤ n 14 3 with the left equality if and only if G ∈ F. Proof. We may assume that G is connected, otherwise we look at each (connected) component separately. Since n ≥ 5, G contains only K2 -cliques and K3 -cliques. Let D be a minimum clique-transversal set of G. We have the following claims. Claim 1. Each component of G[V − D] is a Pi -component, where 1 ≤ i ≤ 3. Let v be a vertex in V −D. If dD (v) = 3, then v is an isolated vertex in G[V −D], so v is a P1 -component of G[V − D]. If dD (v) = 2, then there exists another vertex v of V − D such that v and v are adjacent. Since every clique of G is dominated by some vertex of D, it follows that the edge e = vv lies in at least one K3 -clique and there exists one vertex vD of D that is adjacent to both v and v . So if dD (v ) = 2, then the vertices v, v induce a P2 -component of G[V − D]. If dD (v ) = 1, then there is another vertex v of V − D that is adjacent to v . Clearly, the edge v v is contained in a K3 -clique, and thus v is adjacent to vD as dD (v ) = 1. This implies that the third neighbor of v is distinct from v , and vD belongs to D, so the vertices v, v and v induce a P3 -component of G[V − D]. If dD (v) = 1, following the discussion similar to the case dD (v ) = 1, we can show that there exist two vertices v and v of G[V − D] such that v is adjacent to both v and v and there exist one vertex vD of D such that vD is adjacent to v, v and v . Then the third neighbor of v as well as v distinct from v, vD lies in D, so the vertices v, v and v induce a P3 -component of G[V − D] and Claim 1 follows.
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u e @ @e
u @ @ @e
e @ @e
u @ @ @e
e @ @e
u @ @ @e
u Fig. 1. The graph F1 with τc (F1 ) =
5 |V 14
(F1 )|
Claim 2. For every vertex v of D, dD (v) ≤ 2. The minimality of D implies that v is adjacent to some vertex of V − D for otherwise D − {v} would be a clique-transversal set of G, so Claim 2 follows. First, we present a lower bound on τc (G). Let li be the number of Pi -components of G[V − D] for i = 1, 2, 3. So |V − D| = l1 + 2l2 + 3l3 . By counting the number of edges between D and V − D, we immediately have e[D, V − D] = 3l1 + 4l2 + 5l3 ≤ 3|D|,
(1)
hence |D| ≥ l1 + 43 l2 + 53 l3 . So 9 9 12 |D| ≥ l1 + l2 + 3l3 ≥ |V − D|, 5 5 5
(2)
that is, 9 14 |D| = |D| + |D| ≥ |D| + |V − D| = n. 5 5 5 Consequently, τc (G) = |D| ≥ 14 n. 5 We show next that for a cubic graph G of order n, τc (G) = 14 n if and only if G ∈ F. Suppose G ∈ F, then for some positive integer k, G = Fk and thus |V (G)| = 14k. We choose one vertex of degree 3 in each Ji , together with the 2k new vertices added to J. These vertices clearly form a clique-transversal set of Fk 5 with cardinality 5k, so τc (Fk ) = 14 |V (Fk )|. The darkened vertices of F1 indicated 5 |V (F1 )|. in Fig. 1 form a minimum clique-transversal set of F1 with τc (F1 ) = 14 5 Conversely, suppose that τc (G) = 14 n for a graph G, then there exists some integer k such that n = 14k and let D be its minimum clique-transversal set. Then |D| = 5k. By the above proof, the equalities hold in inequalities (1) and (2). The equality in (1) implies that G[D] has no edges, while the equality in (2) implies that l1 = l2 = 0, and G[V − D] is a collection of P3 -components of cardinality 3k. By the proof of Claim 1, we can see that there is a set of vertices of cardinality 3k of D such that each vertex of the set is precisely adjacent to the three vertices of one P3 -component in G[V − D], which results in 3k copies of J0 . The remainders of D have exactly 2k vertices that are adjacent to the endpoints of all P3 -components in G[V − D]. So G ∈ F.
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Now we present an upper bound on τc (G). By Claim 2, we can partition D into sets D0 = {v ∈ D | dD (v) = 0}, D1 = {v ∈ D | dD (v) = 1} and D2 = {v ∈ D | dD (v) = 2}, and let x, y and z be the cardinality of D0 , D1 and D2 , respectively. Then every vertex of D2 dominates precisely a vertex of V − D. On the other hand, every vertex of V − D is adjacent to at most one vertex of D2 . Otherwise, suppose there exists a vertex v of V − D that is adjacent to both vertices v1 and v2 of D2 , then D ∪ {v} − {v1 , v2 } is a clique-transversal set with order smaller than τc (G), a contradiction. Thus we have |V − D| ≥ |D2 | = z.
(3)
Further, observing that e[D, V − D] = 3x + 2y + z ≤ 3|V − D|, we have 2 1 |V − D| ≥ x + y + z. 3 3
(4)
If z < x + 23 y + 13 z, then z < 32 x + y, so 2n = 2(|D| + |V − D|) 2 1 ≥ 2(x + y + z) + 2(x + y + z) ( by (4)) 3 3 1 = 3(x + y + z) + (3x + y − z) 3 ≥ 3τc (G). Consequently, τc (G) ≤ 23 n. If z ≥ x + 23 y + 13 z, then z ≥ 32 x + y ≥ x + y, so τc (G) = |D| = x + y + z 2 ≤ (x + y + 2z) 3 2 ≤ (|D| + |V − D|) ( by (3)) 3 2 = n, 3 the desired result follows.
If the cubic graph is claw-free, then the upper bound in Theorem 2 can be improved. Theorem 2. If G is a connected claw-free cubic graph of order n, then τc (G) ≤ n 2 and the bound is sharp. Proof. If n ≤ 4, then G = K4 and the result holds. So we may assume that n ≥ 5, thus G contains only the K2 -cliques and K3 -cliques. Let E be the set of edges of all K2 -cliques in G, and V the set of vertices incident with edges of E . We have the following claims. Claim 1. E is a matching in G.
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Otherwise, there exist two edges of E that share a common vertex, say v, and it would yield a claw at v, a contradiction. Claim 2. G[V − V ] consists of only P2 -components. For any vertex v of V − V , there are two K3 -cliques in G containing v for otherwise v would belong to V . Since G is claw-free, it follows that the two cliques share a common edge, say uv, which is incident with v. So u ∈ V − V . But then the other vertices in the two K3 -cliques are contained in V . So both u and v induce a P2 -component of G[V − V ] and Claim 2 follows. By Claims 1 and 2, E ∪ E(G[V − V ]) is exactly a perfect matching in G, so α (G) = n2 . Claim 3. For any vertex v ∈ V , there exactly is a K3 -clique containing v. Otherwise, it must be the case that v ∈ V − V . Claims 2 and 3 imply that for each K3 -clique in G, either its three vertices are from different K2 -cliques in G or it shares a common edge with another K3 -clique. Now we construct a clique-transversal set S of G satisfying Property (A) as follows: (i) For every P2 -component P2 in G[V − V ], |S ∩ P2 | = 1; (ii) for every K2 -clique K2 in G, |S ∩ K2 | ≥ 1; (iii) for every K3 -clique K3 in G, |S ∩ K3 | ≥ 1. We can easily see that S is a clique-transversal set of G. Choose an S such that |S| is minimum. We next show that |S ∩ K2 | = 1 for every K2 -clique K2 in G. Suppose to the contrary that there exist u0 , v1 ∈ S that are in the same K2 -clique in G. By Claim 3, there exists a K3 -clique in G containing v1 . Let {v1 , w1 , u1 } be the set of vertices of the K3 -clique. The minimality of S implies that w1 , u1 ∈ S for otherwise S −{v1 } is a smaller set that satisfies Property (A), a contradiction. Furthermore, w1 and u1 must belong to V . Suppose it is not the case, then w1 and u1 in G[V −V ] would induce a P2 -component, so it follows from (i) that S−{v1 } is a smaller set that satisfies Property (A). Hence w1 , u1 ∈ V −S. Let {u1 , v2 } be the set of vertices of the K2 -clique in G containing u1 . Then v2 must be in S by (ii). By Claim 3 again, we may assume that {v2 , w2 , u2 } is the set of vertices of the K3 -clique in G containing v2 . Then we claim that only the vertex v2 in the K3 -clique belongs to S. Without loss of generality, suppose u2 ∈ S, then we would obtain a new set S = S ∪ {u1 } − {v1 , v2 }. Obviously, S still satisfies Property (A), and we arrive in a contradiction again. So u2 and w2 belong to V − S by (i). Let {u2 , v3 } be the set of vertices of the K2 -clique in G containing u2 . Then v3 ∈ S by (ii) again. We further consider that the K3 clique in G contains v3 . Let {v3 , w3 , u3 } be the set of vertices of the K3 -clique in G containing v3 . Similarly, we can show that v3 ∈ S and w3 , u3 ∈ V − S. We continue the above process by searching alternately for the K2 -cliques and K3 -cliques in G. Finally, we obtain a sequence of K2 -cliques and K3 -cliques alternately occurring in G. In the t-th step, we get a K3 -clique, say K3 (vt , wt , ut ),
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and we claim that only the vertex vt in the K3 -clique belongs to S. Without loss of generality, suppose ut ∈ S. Write the sequence as K2 (u0 , v1 ), K3 (v1 , w1 , u1 ), K2 (u1 , v2 ), K3 (v2 , w2 , u2 ), . . . , K2 (ut−1 , vt ), K3 (vt , wt , ut ).
In the sequence the last K3 -clique contains two vertices of S and the others contain exactly one vertex of S. Let N = {v1 , v2 , . . . , vt }, N = {u1 , u2 , . . . , ut−1 }. According to our construction, we have N ⊆ S and N ∩ S = ∅. Let S = S ∪ N − N . Clearly S satisfies the Property (A) but |S | < |S|, a contradiction. So we can continue the sequence without end, which is impossible as there are a limited number of K3 -cliques in G. So |S ∩ K2 | = 1 for every K2 -clique K2 in G. Therefore, τc (G) ≤ |S| = |V ∩ S| + |(V − V ) ∩ S| = |V |/2 + |V − V |/2 =
n . 2
From the above proof, it is easy to see that if G satisfies V = V , i.e., V −V = ∅, then τc (G) = n2 by Claim 1. Fig. 2 shows an example H1 of a claw-free cubic graph satisfying τc (H1 ) = n2 in which the darkened vertices indicated in Fig. 2 form a minimum clique-transversal set of H1 .
u u @ @e
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u u @ @e
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u p
p
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p
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Fig. 2. The graph H1 with τc (H1 ) =
e @ @e u
n 2
As an immediate consequence of Theorem 2, we have the following result, which shows that for a claw-free cubic graph G the clique-transversal number τc (G) and matching number α1 (G) are comparable. Corollary 1. For any claw-free cubic graph G, τc (G) ≤ α1 (G). In Section 3, however, we give an example H2 shown in Fig. 3 of a cubic graph for which τc (H2 ) is equal to 35 |V (H2 )|. This shows the above result is not true for arbitrary cubic graphs. A graph G is called weakly m-colorable if its vertices can be colored with m colors such that G has no monochromatic cliques. Andreae et al. [3] observed that there is a straightforward relationship between weakly 2-colorable graphs
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and the bound of n/2 for the clique-transversal number, namely τc (G) ≤ n/2 for each weakly 2-colorable graph of order n without isolated vertices, since both color-classes are clique-transversal sets and one of them must have cardinality no larger than n/2. Therefore, we have the following obvious corollary. Corollary 2. Every claw-free cubic graph G is weakly 2-colorable.
3
Conclusion
In Section 2 we could not find an extremal graph G for which τc (G) = 23 n, which means that the upper bound in Theorem 1 may be improved. We close this paper with the following problem. Problem. Is it true that τc (G) ≤ 35 n for cubic graphs ?
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Fig. 3. The graph H2 with τc (H2 ) = 35 n
If it is true, then the upper bound is sharp. It is easy to check that the Peterson Graph attains exactly the upper bound. Furthermore, an example H2 is also shown in Fig. 3. It is not difficult to check that the darkened vertices of H2 indicated in Fig. 3 form a minimum clique-transversal set of H2 .
References 1. Aigner, M., Andreae, T.: Vertex-sets that meet all maximal cliques of a graph, manusript (1986) 2. Andreae, T.: On the clique-transversal number of chordal graphs. Discrete Math. 191, 3–11 (1998) 3. Andreae, T., Schughart, M., Tuza, Z.: Clique-transversal sets of line graphs and complements of line graphs. Discrete Math. 88, 11–20 (1991) 4. Balachandhran, V., Nagavamsi, P., Pandu Rangan, C.: Clique transversal and clique independence on comparability graphs. Inform. Process. Lett. 58, 181–184 (1996) 5. Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Elsevier North Holland, New York (1986)
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6. Chang, M.S., Chen, Y.H., Chang, G.J., Yan, J.H.: Algorithmic aspects of the generalized clique-transversal problem on chordal graphs. Discrete Appl. Math. 66, 189–203 (1996) 7. Chang, G.J., Farber, M., Tuza, Z.: Algorithmic aspects of neighbourhood numbers. SIAM J. Discrete Math. 6, 24–29 (1993) 8. Dahlhaus, E., Mannuel, P.D., Miller, M.: Maximum h-colourable subgraph problem in balanced graphs. Inform. Process. Lett. 65, 301–303 (1998) 9. Dur´ an, G., Lin, M.C., Szwarcfiter, J.L.: On clique-transversals and cliqueindependent sets. Annals of Operations Research 116, 71–77 (2002) 10. Erd˝ os, P., Gallai, T., Tuza, T.: Covering the cliques of a graph with vertices. Discrete Math. 108, 279–289 (1992) 11. Guruswami, V., Pandu Rangan, C.: Algorithmic aspects of clique-transversal and clique-independent sets. Discrete Appl. Math. 100, 183–202 (2000) 12. Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998) 13. Lai, F., Chang, G.J.: An upper bound for the transversal numbers of 4-uniform hypergraphs. J. Combin. Theory Ser. B 50, 129–133 (1990) 14. Lee, C.M., Chang, M.S.: Distance-hereditary graphs are clique-perfect. Discrete Appl. Math. 154, 525–536 (2006) 15. Lonc, Z., Rival, I.: Chains, antichains and fibers. J. Combin. Theory Ser. A 44, 207–228 (1987) 16. Prisner, E.: Graphs with few cliques. In: Alavi, Y., Schwenk, A. (eds.) Graph Theory, Combinatorics and Applications. Proceedings of the 7th Quadrennial International Conference on the Theory and Applications, pp. 945–956. Wiley, Chichester, New York (1995) 17. Tuza, Z.: Covering all cliques of a graph. Discrete Math. 86, 117–126 (1990) 18. Xu, G.J., Shan, E.F., Kang, L.Y., Cheng, T.C.E.: The algorithmic complexity of the minus clique-transversal problem. Appl. Math. Comput. 189, 1410–1418 (2007)
Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China [email protected] 2 Department of Logistics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong [email protected]
Abstract. A clique-transversal set S of a graph G is a set of vertices of G such that S meets all cliques of G. The clique-transversal number, denoted τc (G), is the minimum cardinality of a clique-transversal set in G. In this paper we present an upper bound and a lower bound on τc (G) for cubic graphs, and characterize the extremal cubic graphs achieving the lower bound. In addition, we present a sharp upper bound on τc (G) for claw-free cubic graphs. Keywords: Clique-transversal number; Cubic graph; Claw-free; Bound.
1
Introduction
All graphs considered here are finite, simple and nonempty. For standard graph theory terminology not given here we refer the reader to [5]. Let G = (V, E) be a graph with vertex set V and edge set E. For a vertex v ∈ V , the degree of v is denoted by d(v) and a vertex of degree 0 is said to be an isolated vertex. If d(v) = k for all v ∈ V , then we call G k-regular. In particular, a 3-regular graph is also called a cubic graph. For a subset S ⊆ V , the subgraph induced by S is denoted by G[S], and let dS (v) denote the number of vertices in S that are adjacent to v. For two disjoint subsets T and S of V , write e[T, S] for the number of edges between T and S. The matching number of G is the maximum cardinality among the independent sets of edges of G and is denoted by α1 (G). A perfect matching in G is a matching with the property that every vertex in G is incident with an edge of the matching. A set U ⊆ V is called a vertex cover of G if every edge of G is incident with a vertex in U . The covering number, denoted by α0 (G), is the minimum cardinality of a vertex cover of G. A subset S of V is called a dominating set if every vertex of V − S is adjacent to some vertex in S. The domination
This research was partially supported by The Hong Kong Polytechnic University under grant number G-YX69, the National Nature Science Foundation of China under grant 10571117, the ShuGuang Plan of Shanghai Education Development Foundation under grant 06SG42 and the Development Foundation of Shanghai Education Committee under grant 05AZ04. Corresponding author.
B. Chen, M. Paterson, and G. Zhang (Eds.): ESCAPE 2007, LNCS 4614, pp. 107–115, 2007. c Springer-Verlag Berlin Heidelberg 2007
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number γ(G) of G is the minimum cardinality taken over all dominating sets of G. Domination in graphs has been well studied (see [12]). The concept of the clique-transversal set in graphs can be regarded as a special case of the transversal set in hypergraph theory, which is closely related to domination and seems to have been introduced in [1]. A clique C of a graph G is a complete subgraph maximal under inclusion and |C| ≥ 2. A set D ⊆ V in G is called a clique-transversal set if for every clique C of G, D ∩ V (C) = ∅. The clique-transversal number, denoted τc (G), is the minimum cardinality of a clique-transversal set of G. By definitions, each clique-transversal set in G is clearly a dominating set, so γ(G) ≤ τc (G). To motivate the study of the clique-transversal set in graphs, we present two examples of application where this concept may be used. An application is in terms of communication networks. Consider a graph associated with a communication network where the vertices in the graph correspond to the sites of the network, a clique usually represents a cluster of sites that has the best possible ability to rapidly exchange information among the members of the cluster. The clique-transversal set in the graph is faster to control all clusters and keeps the ability of dominating the whole network. Another application may be found in social networks theory. Every vertex of a graph represents an actor and an edge represents a relationship between two actors. A clique can be viewed as a maximal group of members that have the same property, while a clique-transversal set can be regarded as some kind of organization in the social networks. Then a clique-transversal set claims that each clique in the social networks owns at least one position in this organization. Erd˝ os, Gallai and Tuza [10] observed that the problem of finding a minimum clique-transversal set for an arbitrary graph is NP-hard. Further, it has been proved that the problem is still NP-hard on split graphs (a subclass of chordal graphs) [7], cocomparability, planar, line and total graphs [11], undirected path graphs, and k-trees with unbounded k [6]. However, there are polynomial time algorithms to find τc for comparability graphs [4], strongly chordal graphs [7], Helly circular-arc graphs [16], and distance-hereditary graphs [14]. In [10], Erd˝ os et al. investigated the bounds on τc , and√showed that every graph of order n has clique-transversal number at most n − 2n + 32 , and if all cliques are relatively large, then a sightly better upper bound can be obtained. However, they also observed that τc (G) can be very close to n = |V (G)|, namely τc = n − o(n) can hold. It is interesting to note that τc drastically decreases when some assumptions are put on the graph G. From this point of view, Tuza [17] and Andreae [2] established upper bounds on τc for chordal graphs. Andreae et al. [3] studied classes of graphs G of order n for which τc (G) ≤ n/2. They showed that (i) all connected line graphs with the exception of odd cycles, and (ii) all complements of line graphs with the exception of five small graphs have clique-transversal numbers at most one-half their orders. For other investigations on the cliquetransversal number of graphs, we refer the reader to [6, 8, 9, 13, 15, 18]. In this paper we shall focus on cubic graphs. We show that if G is any cubic graph, then 5n/14 ≤ τc (G) ≤ 2n/3, and the extremal graphs attaining the lower
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bound are characterized. Also, we show that a claw-free cubic graph has cliquetransversal number at most one-half its order, and the upper bound is sharp.
2
Clique-Transversal Number in Cubic Graphs
In this section we give lower and upper bounds on the clique-transversal number of a cubic graph in terms of its order and we characterize the graph attaining the lower bound. For this purpose, we define a family F of graphs as follows: For each integer k ≥ 1, let J0 be the graph obtained from a complete graph K4 on four vertices by deleting one edge, and let J be the disjoint union of J1 , J2 , . . . , J3k of 3k copies of J0 . Let Fk be a family of cubic graphs obtained from J by adding 2k new vertices and 6k edges that join each new vertex exactly to three vertices of degree 2 of J so that each vertex has degree 3. Let F = {Fk | k ≥ 1}. The graph F1 is shown in Fig. 1. For notational convenience, every clique of order m of a graph G is called a Km -clique of G, and a component of G is called a H-component if it is isomorphic to a given graph H. Theorem 1. If G is a cubic graph of order n ≥ 5, then 2 5 n ≤ τc (G) ≤ n 14 3 with the left equality if and only if G ∈ F. Proof. We may assume that G is connected, otherwise we look at each (connected) component separately. Since n ≥ 5, G contains only K2 -cliques and K3 -cliques. Let D be a minimum clique-transversal set of G. We have the following claims. Claim 1. Each component of G[V − D] is a Pi -component, where 1 ≤ i ≤ 3. Let v be a vertex in V −D. If dD (v) = 3, then v is an isolated vertex in G[V −D], so v is a P1 -component of G[V − D]. If dD (v) = 2, then there exists another vertex v of V − D such that v and v are adjacent. Since every clique of G is dominated by some vertex of D, it follows that the edge e = vv lies in at least one K3 -clique and there exists one vertex vD of D that is adjacent to both v and v . So if dD (v ) = 2, then the vertices v, v induce a P2 -component of G[V − D]. If dD (v ) = 1, then there is another vertex v of V − D that is adjacent to v . Clearly, the edge v v is contained in a K3 -clique, and thus v is adjacent to vD as dD (v ) = 1. This implies that the third neighbor of v is distinct from v , and vD belongs to D, so the vertices v, v and v induce a P3 -component of G[V − D]. If dD (v) = 1, following the discussion similar to the case dD (v ) = 1, we can show that there exist two vertices v and v of G[V − D] such that v is adjacent to both v and v and there exist one vertex vD of D such that vD is adjacent to v, v and v . Then the third neighbor of v as well as v distinct from v, vD lies in D, so the vertices v, v and v induce a P3 -component of G[V − D] and Claim 1 follows.
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u Fig. 1. The graph F1 with τc (F1 ) =
5 |V 14
(F1 )|
Claim 2. For every vertex v of D, dD (v) ≤ 2. The minimality of D implies that v is adjacent to some vertex of V − D for otherwise D − {v} would be a clique-transversal set of G, so Claim 2 follows. First, we present a lower bound on τc (G). Let li be the number of Pi -components of G[V − D] for i = 1, 2, 3. So |V − D| = l1 + 2l2 + 3l3 . By counting the number of edges between D and V − D, we immediately have e[D, V − D] = 3l1 + 4l2 + 5l3 ≤ 3|D|,
(1)
hence |D| ≥ l1 + 43 l2 + 53 l3 . So 9 9 12 |D| ≥ l1 + l2 + 3l3 ≥ |V − D|, 5 5 5
(2)
that is, 9 14 |D| = |D| + |D| ≥ |D| + |V − D| = n. 5 5 5 Consequently, τc (G) = |D| ≥ 14 n. 5 We show next that for a cubic graph G of order n, τc (G) = 14 n if and only if G ∈ F. Suppose G ∈ F, then for some positive integer k, G = Fk and thus |V (G)| = 14k. We choose one vertex of degree 3 in each Ji , together with the 2k new vertices added to J. These vertices clearly form a clique-transversal set of Fk 5 with cardinality 5k, so τc (Fk ) = 14 |V (Fk )|. The darkened vertices of F1 indicated 5 |V (F1 )|. in Fig. 1 form a minimum clique-transversal set of F1 with τc (F1 ) = 14 5 Conversely, suppose that τc (G) = 14 n for a graph G, then there exists some integer k such that n = 14k and let D be its minimum clique-transversal set. Then |D| = 5k. By the above proof, the equalities hold in inequalities (1) and (2). The equality in (1) implies that G[D] has no edges, while the equality in (2) implies that l1 = l2 = 0, and G[V − D] is a collection of P3 -components of cardinality 3k. By the proof of Claim 1, we can see that there is a set of vertices of cardinality 3k of D such that each vertex of the set is precisely adjacent to the three vertices of one P3 -component in G[V − D], which results in 3k copies of J0 . The remainders of D have exactly 2k vertices that are adjacent to the endpoints of all P3 -components in G[V − D]. So G ∈ F.
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Now we present an upper bound on τc (G). By Claim 2, we can partition D into sets D0 = {v ∈ D | dD (v) = 0}, D1 = {v ∈ D | dD (v) = 1} and D2 = {v ∈ D | dD (v) = 2}, and let x, y and z be the cardinality of D0 , D1 and D2 , respectively. Then every vertex of D2 dominates precisely a vertex of V − D. On the other hand, every vertex of V − D is adjacent to at most one vertex of D2 . Otherwise, suppose there exists a vertex v of V − D that is adjacent to both vertices v1 and v2 of D2 , then D ∪ {v} − {v1 , v2 } is a clique-transversal set with order smaller than τc (G), a contradiction. Thus we have |V − D| ≥ |D2 | = z.
(3)
Further, observing that e[D, V − D] = 3x + 2y + z ≤ 3|V − D|, we have 2 1 |V − D| ≥ x + y + z. 3 3
(4)
If z < x + 23 y + 13 z, then z < 32 x + y, so 2n = 2(|D| + |V − D|) 2 1 ≥ 2(x + y + z) + 2(x + y + z) ( by (4)) 3 3 1 = 3(x + y + z) + (3x + y − z) 3 ≥ 3τc (G). Consequently, τc (G) ≤ 23 n. If z ≥ x + 23 y + 13 z, then z ≥ 32 x + y ≥ x + y, so τc (G) = |D| = x + y + z 2 ≤ (x + y + 2z) 3 2 ≤ (|D| + |V − D|) ( by (3)) 3 2 = n, 3 the desired result follows.
If the cubic graph is claw-free, then the upper bound in Theorem 2 can be improved. Theorem 2. If G is a connected claw-free cubic graph of order n, then τc (G) ≤ n 2 and the bound is sharp. Proof. If n ≤ 4, then G = K4 and the result holds. So we may assume that n ≥ 5, thus G contains only the K2 -cliques and K3 -cliques. Let E be the set of edges of all K2 -cliques in G, and V the set of vertices incident with edges of E . We have the following claims. Claim 1. E is a matching in G.
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Otherwise, there exist two edges of E that share a common vertex, say v, and it would yield a claw at v, a contradiction. Claim 2. G[V − V ] consists of only P2 -components. For any vertex v of V − V , there are two K3 -cliques in G containing v for otherwise v would belong to V . Since G is claw-free, it follows that the two cliques share a common edge, say uv, which is incident with v. So u ∈ V − V . But then the other vertices in the two K3 -cliques are contained in V . So both u and v induce a P2 -component of G[V − V ] and Claim 2 follows. By Claims 1 and 2, E ∪ E(G[V − V ]) is exactly a perfect matching in G, so α (G) = n2 . Claim 3. For any vertex v ∈ V , there exactly is a K3 -clique containing v. Otherwise, it must be the case that v ∈ V − V . Claims 2 and 3 imply that for each K3 -clique in G, either its three vertices are from different K2 -cliques in G or it shares a common edge with another K3 -clique. Now we construct a clique-transversal set S of G satisfying Property (A) as follows: (i) For every P2 -component P2 in G[V − V ], |S ∩ P2 | = 1; (ii) for every K2 -clique K2 in G, |S ∩ K2 | ≥ 1; (iii) for every K3 -clique K3 in G, |S ∩ K3 | ≥ 1. We can easily see that S is a clique-transversal set of G. Choose an S such that |S| is minimum. We next show that |S ∩ K2 | = 1 for every K2 -clique K2 in G. Suppose to the contrary that there exist u0 , v1 ∈ S that are in the same K2 -clique in G. By Claim 3, there exists a K3 -clique in G containing v1 . Let {v1 , w1 , u1 } be the set of vertices of the K3 -clique. The minimality of S implies that w1 , u1 ∈ S for otherwise S −{v1 } is a smaller set that satisfies Property (A), a contradiction. Furthermore, w1 and u1 must belong to V . Suppose it is not the case, then w1 and u1 in G[V −V ] would induce a P2 -component, so it follows from (i) that S−{v1 } is a smaller set that satisfies Property (A). Hence w1 , u1 ∈ V −S. Let {u1 , v2 } be the set of vertices of the K2 -clique in G containing u1 . Then v2 must be in S by (ii). By Claim 3 again, we may assume that {v2 , w2 , u2 } is the set of vertices of the K3 -clique in G containing v2 . Then we claim that only the vertex v2 in the K3 -clique belongs to S. Without loss of generality, suppose u2 ∈ S, then we would obtain a new set S = S ∪ {u1 } − {v1 , v2 }. Obviously, S still satisfies Property (A), and we arrive in a contradiction again. So u2 and w2 belong to V − S by (i). Let {u2 , v3 } be the set of vertices of the K2 -clique in G containing u2 . Then v3 ∈ S by (ii) again. We further consider that the K3 clique in G contains v3 . Let {v3 , w3 , u3 } be the set of vertices of the K3 -clique in G containing v3 . Similarly, we can show that v3 ∈ S and w3 , u3 ∈ V − S. We continue the above process by searching alternately for the K2 -cliques and K3 -cliques in G. Finally, we obtain a sequence of K2 -cliques and K3 -cliques alternately occurring in G. In the t-th step, we get a K3 -clique, say K3 (vt , wt , ut ),
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and we claim that only the vertex vt in the K3 -clique belongs to S. Without loss of generality, suppose ut ∈ S. Write the sequence as K2 (u0 , v1 ), K3 (v1 , w1 , u1 ), K2 (u1 , v2 ), K3 (v2 , w2 , u2 ), . . . , K2 (ut−1 , vt ), K3 (vt , wt , ut ).
In the sequence the last K3 -clique contains two vertices of S and the others contain exactly one vertex of S. Let N = {v1 , v2 , . . . , vt }, N = {u1 , u2 , . . . , ut−1 }. According to our construction, we have N ⊆ S and N ∩ S = ∅. Let S = S ∪ N − N . Clearly S satisfies the Property (A) but |S | < |S|, a contradiction. So we can continue the sequence without end, which is impossible as there are a limited number of K3 -cliques in G. So |S ∩ K2 | = 1 for every K2 -clique K2 in G. Therefore, τc (G) ≤ |S| = |V ∩ S| + |(V − V ) ∩ S| = |V |/2 + |V − V |/2 =
n . 2
From the above proof, it is easy to see that if G satisfies V = V , i.e., V −V = ∅, then τc (G) = n2 by Claim 1. Fig. 2 shows an example H1 of a claw-free cubic graph satisfying τc (H1 ) = n2 in which the darkened vertices indicated in Fig. 2 form a minimum clique-transversal set of H1 .
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Fig. 2. The graph H1 with τc (H1 ) =
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n 2
As an immediate consequence of Theorem 2, we have the following result, which shows that for a claw-free cubic graph G the clique-transversal number τc (G) and matching number α1 (G) are comparable. Corollary 1. For any claw-free cubic graph G, τc (G) ≤ α1 (G). In Section 3, however, we give an example H2 shown in Fig. 3 of a cubic graph for which τc (H2 ) is equal to 35 |V (H2 )|. This shows the above result is not true for arbitrary cubic graphs. A graph G is called weakly m-colorable if its vertices can be colored with m colors such that G has no monochromatic cliques. Andreae et al. [3] observed that there is a straightforward relationship between weakly 2-colorable graphs
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and the bound of n/2 for the clique-transversal number, namely τc (G) ≤ n/2 for each weakly 2-colorable graph of order n without isolated vertices, since both color-classes are clique-transversal sets and one of them must have cardinality no larger than n/2. Therefore, we have the following obvious corollary. Corollary 2. Every claw-free cubic graph G is weakly 2-colorable.
3
Conclusion
In Section 2 we could not find an extremal graph G for which τc (G) = 23 n, which means that the upper bound in Theorem 1 may be improved. We close this paper with the following problem. Problem. Is it true that τc (G) ≤ 35 n for cubic graphs ?
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Fig. 3. The graph H2 with τc (H2 ) = 35 n
If it is true, then the upper bound is sharp. It is easy to check that the Peterson Graph attains exactly the upper bound. Furthermore, an example H2 is also shown in Fig. 3. It is not difficult to check that the darkened vertices of H2 indicated in Fig. 3 form a minimum clique-transversal set of H2 .
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